L(s) = 1 | + i·2-s + (−0.707 + 0.707i)3-s − 4-s + (2.05 − 0.891i)5-s + (−0.707 − 0.707i)6-s + (−1.56 + 1.56i)7-s − i·8-s − 1.00i·9-s + (0.891 + 2.05i)10-s + 3.92i·11-s + (0.707 − 0.707i)12-s − 0.319i·13-s + (−1.56 − 1.56i)14-s + (−0.819 + 2.08i)15-s + 16-s + 1.25·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (0.917 − 0.398i)5-s + (−0.288 − 0.288i)6-s + (−0.591 + 0.591i)7-s − 0.353i·8-s − 0.333i·9-s + (0.282 + 0.648i)10-s + 1.18i·11-s + (0.204 − 0.204i)12-s − 0.0885i·13-s + (−0.418 − 0.418i)14-s + (−0.211 + 0.537i)15-s + 0.250·16-s + 0.304·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.039897715\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.039897715\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.05 + 0.891i)T \) |
| 37 | \( 1 + (-0.0100 - 6.08i)T \) |
good | 7 | \( 1 + (1.56 - 1.56i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.92iT - 11T^{2} \) |
| 13 | \( 1 + 0.319iT - 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 + (2.17 - 2.17i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.15iT - 23T^{2} \) |
| 29 | \( 1 + (-1.22 - 1.22i)T + 29iT^{2} \) |
| 31 | \( 1 + (6.78 - 6.78i)T - 31iT^{2} \) |
| 41 | \( 1 - 3.92iT - 41T^{2} \) |
| 43 | \( 1 + 1.87iT - 43T^{2} \) |
| 47 | \( 1 + (4.88 - 4.88i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.03 + 3.03i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.70 - 4.70i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.56 - 5.56i)T - 61iT^{2} \) |
| 67 | \( 1 + (6.88 + 6.88i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + (-8.53 + 8.53i)T - 73iT^{2} \) |
| 79 | \( 1 + (-3.59 + 3.59i)T - 79iT^{2} \) |
| 83 | \( 1 + (1.13 + 1.13i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.33 + 1.33i)T + 89iT^{2} \) |
| 97 | \( 1 - 4.01T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04829122103855728976179671309, −9.354795707155813439088109657328, −8.797954060141284018044016989188, −7.68209483785444186980593663046, −6.59789128313644509884574874464, −6.07430972165039190893437264406, −5.15166860865391047193251242424, −4.53397288699849762950939953407, −3.12910842690529033409904915497, −1.66364095855178720391674128002,
0.47713989622159454876859585092, 1.87723683859071734073015142074, 2.99272026340767367794082738993, 3.93841277591278239286473546813, 5.31509736677735945732380111685, 6.02888379050046764690095448331, 6.80675493499844966748002892588, 7.78308991951030774666217833546, 8.908796939744377554103499450438, 9.597117257676220963967875911477